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Let $F$ be a field and $f: \mathbb{Z} \to F$ be a ring epimorphism.
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i would like to understand what is a difference between parallel and orthogonal projection?let us consider following picture

we have two non othogonal basis and vector A with coordinates($7$,$2$),i would like to find parallel projection of this vector to these basis,i am studying Covariant and Contrivant components,so i would like to understand how to find parallel projection and also orthogonal projection?according to wikipedia,Orthogonal projection is defined by

http://en.wikipedia.org/wiki/Vector_projection

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what about parallel projection?please help me

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In geometric terms …

In a parallel projection, points are projected (onto some plane) in a direction that is parallel to some fixed given vector.

In an orthogonal projection, points are projected (onto some plane) in a direction that is normal to the plane.

So, all orthogonal projections are parallel projections, but not vice versa. A parallel projection that is not an orthogonal projection is called an “oblique” projection.

This could all be translated into the language of linear algebra, I suppose, but I don’t think that would make it any clearer.

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